Problem: $\int (- x^3 -8 x +5)\,dx=$ $+C$
Solution: We can use the sum rule and the constant multiple rule for indefinite integrals: $\begin{aligned} &\int [f(x)+g(x)]dx=\int f(x)\,dx+\int g(x)\,dx \\\\\\ &\int k\cdot f(x)= k\cdot\int f(x)\,dx \end{aligned}$ Using the sum and the constant multiple rules, we can rewrite our integral as follows: $\int (- x^3 -8 x +5)\,dx= -1\int x^3\,dx -8\int x\,dx +5\int 1\,dx$ Now we can find each indefinite integral using the reverse power rule: $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ Note: we can only use the reverse power rule because $n \neq -1$. $\begin{aligned} &\phantom{=}\int (- x^3 -8 x +5)\,dx \\\\ &= -\int x^3\,dx -8\int x\,dx +5\int 1\,dx \\\\ &=- \dfrac{x^4}{4} -8\dfrac{x^2}{2} +5\dfrac{x^1}{1}+C \\\\ &=-\dfrac{1}{4} x^4 -4 x^2 +5 x+C \end{aligned}$ In conclusion, $\int (- x^3 -8 x +5)\,dx=-\dfrac{1}{4} x^4 -4 x^2 +5 x+C$